Off-shell \(\mathcal {N}=1\)  supersymmetric extensions of the new topologically massive higher-spin gauge models (REF ) were also proposed in [1]}. For any integer \(n\ge 1\) , the gauge-invariant action corresponding to the unconstrained prepotential \(H_{\alpha (n)}\)  may be recast into the form
\(\mathbb {S}_{\text{NTM}}^{(n)}[H]=- \frac{{\rm i}^n}{2^{\left\lfloor {n/2}\right\rfloor +1}}\frac{1}{M}\int {\rm d}^{3|2}z \, E \,\mathfrak {W}^{\alpha (n)}(H) \big (\mathbb {F}-M\big )H_{\alpha (n)}~. \) 
